We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. allowing zero as some Ouvry and Polychronakos 2019. We give bijection between these two varieties of determine various combinatorial properties multicompositions. In particular, we the count multicompositions number all parts, positive zeros. Then, working from three types r...